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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/>

<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E10">A1</font></i>: 
( the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="bhsp_3.html#NR5" title="BHSP_3:NR.5">is_Hilbert_space</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c2">S</font> be    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">H</font> be   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E2:9"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">H</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">x</font> ) )
 ;<br/>

<div><i><font color="Green" title="E12">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="bhsp_6.html#V1" title="BHSP_6:attr.1">summable_set</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green" title="E14">A5</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font>
 ;<br/><b>set </b><font color="Maroon" title="c5">e'</font> =  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">e</font>;<br/><a NAME="E2:9_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="square_1.html#T93" title="SQUARE_1:th.93">SQUARE_1:93</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">e1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green" title="E15">A6</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font> &amp; <font color="Maroon" title="c6">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">e</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T1" title="CHAIN_1:th.1">CHAIN_1:1</a></i>;<br/><a NAME="E5:9_1_1"/>
<font color="Maroon" title="c6">e1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E4:9_1_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/><a NAME="E6:9_1_1"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c6">e1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c6">e1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="bhsp_7.html#E4:9_1_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">Y0</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:9_1_1"/><i><font color="Green" title="E17">A8</font></i>: 
( not <font color="Maroon" title="c7">Y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b1">Y1</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b1">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c7">Y0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Y1</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Olive" title="b1">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font> ) )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="bhsp_7.html#E1:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="bhsp_7.html#E4:9_1_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="bhsp_6.html#T10" title="BHSP_6:th.10">BHSP_6:10</a></i>;<br/><b>take </b><font color="Maroon" title="c8">Y0</font> = <font color="Maroon" title="c7">Y0</font>;<br/><b>thus </b><a NAME="E9:9_1_1"/>
( not <font color="Maroon" title="c8">Y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E8:9_1_1"><i><font color="Green" title="E17">A8</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c9">Y1</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E10:9_1_1"/><i><font color="Green" title="E18">A9</font></i>: 
( not <font color="Maroon" title="c9">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c9">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c8">Y0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Y1</font> )
 ;<br/><a NAME="E11:9_1_1"/><i><font color="Green" title="E19">A10</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#T34" title="BHSP_1:th.34">BHSP_1:34</a></i>;<br/><a NAME="E12:9_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E8:9_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="bhsp_7.html#E10:9_1_1"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/><a NAME="E13:9_1_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E11:9_1_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/><a NAME="E14:9_1_1"/><b>then </b><i><font color="Green" title="E20">A11</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E6:9_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:9_1_1"/>
<font color="Maroon" title="c9">Y1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E10:9_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="bhsp_7.html#T5" target="_self" title="BHSP_7:th.5">Th5</a></i>;<br/><a NAME="E16:9_1_1"/><b>then </b><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c9">Y1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M1" title="BHSP_5:mode.1">OrthogonalFamily</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bhsp_5.html#D9" title="BHSP_5:def.9">BHSP_5:def 9</a></i>;<br/><a NAME="E17:9_1_1"/><i><font color="Green" title="E22">A13</font></i>: 
<font color="Maroon" title="c9">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="bhsp_7.html#E10:9_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:9_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y1</font> holds <br/><font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="bhsp_7.html#E10:9_1_1"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/><a NAME="E19:9_1_1"/><b>then </b><i><font color="Green" title="E23">A14</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c9">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="bhsp_7.html#E10:9_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="bhsp_7.html#E16:9_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="bhsp_7.html#E17:9_1_1"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="bhsp_7.html#T3" target="_self" title="BHSP_7:th.3">Th3</a></i>;<br/><a NAME="E20:9_1_1"/><i><font color="Green" title="E24">A15</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4" title="BHSP_1:def.4">BHSP_1:def 4</a></i>;<br/><a NAME="E21:9_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/><a NAME="E22:9_1_1"/><b>then </b><i><font color="Green" title="E25">A16</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c9">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c9">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E19:9_1_1"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="bhsp_7.html#E20:9_1_1"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E23:9_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c9">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E19:9_1_1"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/><b>hence </b><a NAME="E24:9_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c9">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E14:9_1_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="bhsp_7.html#E22:9_1_1"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:9_1"/>
<font color="Maroon" title="c2">S</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="bhsp_7.html#T1" target="_self" title="BHSP_7:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_2"/><i><font color="Green" title="E13">A17</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Maroon" title="c3">H</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:9_2_1"/><i><font color="Green" title="E14">A18</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font>
 ;<br/><b>set </b><font color="Maroon" title="c5">e1</font> = <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font>;<br/><a NAME="E2:9_2_1"/>
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font> )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9_2_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">Y0</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:9_2_1"/><i><font color="Green" title="E15">A19</font></i>: 
( not <font color="Maroon" title="c6">Y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c6">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b1">Y1</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b1">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">Y0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Y1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Olive" title="b1">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font> ) )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9_2"><i><font color="Green" title="E13">A17</font></i></a>, <a class="ref" href="bhsp_7.html#T1" target="_self" title="BHSP_7:th.1">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">Y1</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:9_2_1_1"/><i><font color="Green" title="E16">A20</font></i>: 
( not <font color="Maroon" title="c7">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">Y0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Y1</font> )
 ;<br/><b>set </b><font color="Maroon" title="c8">F</font> =  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> ;<br/><a NAME="E2:9_2_1_1"/><i><font color="Green" title="E17">A21</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E4:9_2_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="txt" href="bhsp_7.html#E1:9_2_1_1"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><a NAME="E3:9_2_1_1"/>
 <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></i>;<br/><a NAME="E4:9_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E2:9_2_1_1"><i><font color="Green" title="E17">A21</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E5:9_2_1_1"/><b>then </b><i><font color="Green" title="E18">A22</font></i>: 
 <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E6:9_2_1_1"/>
<font color="Maroon" title="c7">Y1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9_2_1_1"><i><font color="Green" title="E16">A20</font></i></a>, <a class="ref" href="bhsp_7.html#T5" target="_self" title="BHSP_7:th.5">Th5</a></i>;<br/><a NAME="E7:9_2_1_1"/><b>then </b><i><font color="Green" title="E19">A23</font></i>: 
<font color="Maroon" title="c7">Y1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M1" title="BHSP_5:mode.1">OrthogonalFamily</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bhsp_5.html#D9" title="BHSP_5:def.9">BHSP_5:def 9</a></i>;<br/><a NAME="E8:9_2_1_1"/><i><font color="Green" title="E20">A24</font></i>: 
<font color="Maroon" title="c7">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="bhsp_7.html#E1:9_2_1_1"><i><font color="Green" title="E16">A20</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:9_2_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y1</font> holds <br/><font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E2:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="bhsp_7.html#E1:9_2_1_1"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><a NAME="E10:9_2_1_1"/><b>then </b><i><font color="Green" title="E21">A25</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c7">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="bhsp_7.html#E1:9_2_1_1"><i><font color="Green" title="E16">A20</font></i></a>, <a class="txt" href="bhsp_7.html#E7:9_2_1_1"><i><font color="Green" title="E19">A23</font></i></a>, <a class="txt" href="bhsp_7.html#E8:9_2_1_1"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="bhsp_7.html#T3" target="_self" title="BHSP_7:th.3">Th3</a></i>;<br/><a NAME="E11:9_2_1_1"/><i><font color="Green" title="E22">A26</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></i>;<br/><a NAME="E12:9_2_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4" title="BHSP_1:def.4">BHSP_1:def 4</a></i>;<br/><a NAME="E13:9_2_1_1"/><b>then </b><i><font color="Green" title="E23">A27</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E5:9_2_1_1"><i><font color="Green" title="E18">A22</font></i></a>, <a class="txt" href="bhsp_7.html#E10:9_2_1_1"><i><font color="Green" title="E21">A25</font></i></a>, <a class="txt" href="bhsp_7.html#E11:9_2_1_1"><i><font color="Green" title="E22">A26</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E14:9_2_1_1"/><i><font color="Green" title="E24">A28</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#T34" title="BHSP_1:th.34">BHSP_1:34</a></i>;<br/><a NAME="E15:9_2_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/><a NAME="E16:9_2_1_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E13:9_2_1_1"><i><font color="Green" title="E23">A27</font></i></a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a></i>;<br/><a NAME="E17:9_2_1_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9_2_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>;<br/><b>hence </b><a NAME="E18:9_2_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c7">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E14:9_2_1_1"><i><font color="Green" title="E24">A28</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:9_2_1"/>
 ex <font color="Olive" title="b1">Y0</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( not <font color="Olive" title="b1">Y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b2">Y1</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b2">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b2">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">Y0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Y1</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Olive" title="b2">Y1</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">e</font> ) )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E4:9_2_1"><i><font color="Green" title="E15">A19</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:9_2"/>
<font color="Maroon" title="c2">S</font> is <a href="bhsp_6.html#V1" title="BHSP_6:attr.1">summable_set</a>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="bhsp_6.html#T10" title="BHSP_6:th.10">BHSP_6:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:9"/>
( <font color="Maroon" title="c2">S</font> is <a href="bhsp_6.html#V1" title="BHSP_6:attr.1">summable_set</a> iff <font color="Maroon" title="c2">S</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E3:9"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>


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